Question

a. In the absorption spectrum of the complex ion \(\mathrm{Cr}(\mathrm{NCS})_{6}^{3-}\) there is a band corresponding to the absorption of a photon of light with an energy of \(1.75 \times 10^{4} \mathrm{cm}^{-1}\). Given 1 \(\mathrm{cm}^{-1}=1.986 \times 10^{-23} \mathrm{J},\) what is the wavelength of this photon? b. The \(\mathrm{Cr}-\mathrm{N}-\mathrm{C}\) bond angle in \(\mathrm{Cr}(\mathrm{NCS})_{6}^{3-}\) is predicted to be \(180^{\circ} .\) What is the hybridization of the \(\mathrm{N}\) atom in the NCS - ligand when a Lewis acid-base reaction occurs between \(\mathrm{Cr}^{3+}\) and \(\mathrm{NCS}^{-}\) that would give a \(180^{\circ}\) \(\mathrm{Cr}-\mathrm{N}-\mathrm{C}\) bond angle? \(\mathrm{Cr}(\mathrm{NCS})_{6}^{3-}\) undergoes substitution by ethylenediamine (en) according to the equation \(\mathrm{Cr}(\mathrm{NCS})_{6}^{3-}+2 \mathrm{en} \longrightarrow \mathrm{Cr}(\mathrm{NCS})_{2}(\mathrm{en})_{2}^{+}+4 \mathrm{NCS}^{-}\) Does \(\operatorname{Cr}(\mathrm{NCS})_{2}(\mathrm{en})_{2}^{+}\) exhibit geometric isomerism? Does \(\mathrm{Cr}(\mathrm{NCS})_{2}(\mathrm{en})_{2}^{+}\) exhibit optical isomerism?

Short answer

The wavelength of the photon is approximately \(571\: \mathrm{nm}\). The hybridization of the N atom in the NCS- ligand is sp. The complex \(\mathrm{Cr}(\mathrm{NCS})_{2}(\mathrm{en})_{2}^{+}\) exhibits geometric isomerism, with possible arrangements being trans and cis isomers. The complex can exhibit optical isomerism, depending on the arrangement of the ligands; while the trans isomer does not exhibit optical isomerism, the cis isomer does.

Step-by-step solution

Find the Wavelength of the Photon

The energy of a photon can be calculated using the formula: \(E_{photon} = \dfrac{hc}{\lambda}\), where h is the Planck's constant, c is the speed of light, and \(\lambda\) is the wavelength of the photon. Given the energy per cm and a conversion factor for Joules, we can find the wavelength. Step 1: Convert the given energy to Joules. Given energy: \(1.75 \times 10^4\: \mathrm{cm}^{-1}\), Conversion factor: \(1\:\mathrm{cm}^{-1} = 1.986 \times 10^{-23} \:\mathrm{J}\), Energy in Joules: \((1.75 \times 10^4 \:\mathrm{cm}^{-1}) \times (1.986 \times 10^{-23} \:\mathrm{J}\:\mathrm{cm}) = 3.475 \times 10^{-19} \mathrm{J}\) Step 2: Calculate the wavelength. Using the energy-wavelength relationship: \(\lambda = \dfrac{hc}{E_{photon}}\), where h = \(6.626 \times 10^{-34}\: \mathrm{Js}\) and c = \(3.0 \times 10^8\: \mathrm{m/s}\), \(\lambda = \dfrac{(6.626 \times 10^{-34}\: \mathrm{Js})(3.0 \times 10^8\: \mathrm{m/s})}{3.475 \times 10^{-19} \mathrm{J}} \approx 5.71 \times 10^{-7} \:\mathrm{m}\) or \(571 \:\mathrm{nm}\) The wavelength of the photon is approximately \(571\: \mathrm{nm}\).

Determine the Hybridization of N Atom

To obtain a \(180^{\circ}\) Cr-N-C bond angle, the hybridization of the N atom in the NCS- ligand should result in a linear geometry around the N atom. The required hybridization is sp.

Analyze Geometric Isomerism

Geometric isomerism occurs when there are different arrangements of ligands around a central metal atom. In the complex \(\rm{Cr(NCS)_2(en)_2^{+}}\), there are two NCS- ligands and two en ligands around the Cr atom. The possible arrangements for these ligands are trans and cis isomers. Hence, the complex exhibits geometric isomerism.

Analyze Optical Isomerism

Optical isomerism occurs when a compound has a non-superimposable mirror image, which means it cannot be laid on top of its mirror image and have all their atoms match up. In this case, the trans isomer of \(\rm{Cr(NCS)_2(en)_2^{+}}\) has a plane of symmetry and does not exhibit optical isomerism. However, the cis isomer does not have a plane of symmetry, thus it can exhibit optical isomerism. Therefore, the complex can exhibit optical isomerism, depending on the arrangement of the ligands.

Key concepts

Absorption Spectrum

In chemistry, the absorption spectrum is a critical concept, especially in understanding coordination complexes. It shows the range of wavelengths (or energies) that a substance can absorb. When a compound, like a coordination complex, absorbs specific wavelengths of light, it triggers electronic transitions. These transitions occur when electrons move between energy levels.For a coordination complex such as \(\mathrm{Cr}( ext{NCS})_{6}^{3-}\), the absorption spectrum tells us about the light energy it absorbs. This is associated with the electronic transitions within the metal center. Coordination complexes typically exhibit colors due to these transitions. These colors correlate with the absorption of photons in particular regions of the visible spectrum.Key points about the absorption spectrum:

• It provides insight into the energy levels present in a compound.
• It can help identify and characterize complex ions and their properties.
• Analyzing the absorption spectrum can reveal ligands' effects and the coordination environment around the metal center.

Photon Wavelength Calculation

Calculating the photon wavelength is a skill that provides insights into the energy interactions within molecules. The energy of a photon (\(E_{\text{photon}}\) ) can be expressed using the formula: \(E_{\text{photon}} = \dfrac{hc}{\lambda}\), where:- \(h\) is the Planck's constant \( (6.626 \times 10^{-34} \, \text{Js}) \),- \(c\) is the speed of light \( (3.0 \times 10^{8} \, \text{m/s}) \), and- \(\lambda\) is the wavelength.By rearranging this to \(\lambda = \dfrac{hc}{E_{\text{photon}}}\), we can find the wavelength when the energy is given. In our exercise with \(\mathrm{Cr}( ext{NCS})_{6}^{3-}\), we calculated the energy in Joules and then found the wavelength, which was about \(571 \, \text{nm}\).Remember, the wavelength tells us the type of electromagnetic radiation that has an equivalent energy. In our case, the absorbed light falls into the visible region, creating the color observed in the complex.

Geometric Isomerism

Geometric isomerism is an intriguing phenomenon in coordination complexes. It involves different spatial arrangements of ligands around a central metal atom, leading to distinct isomers. This is commonly seen in complexes with square planar or octahedral geometries.For the complex \(\mathrm{Cr}( ext{NCS})_{2}( ext{en})_{2}^{+}\), there are two NCS- ligands and two en ligands. Because geometric isomers can form, these ligands can arrange differently:- **Cis Isomer**: Here, the same type of ligand is adjacent.- **Trans Isomer**: Here, the same type of ligand is opposite to each other.The ability to form these isomers influences the properties of the complex, such as its reactivity and color. Geometric isomerism illustrates how the coordination sphere's arrangement can affect a complex's chemistry, which is why it's a critical concept in understanding coordination compounds.

Optical Isomerism

Optical isomerism is when coordination complexes exhibit chirality, displaying isomers that are mirror images of each other, yet non-superimposable. This property is much like left and right human hands.For the complex \(\mathrm{Cr}( ext{NCS})_{2}( ext{en})_{2}^{+}\), optical isomerism exists due to the potential formation of asymmetric arrangements. Notably, the cis isomer lacks a plane of symmetry and can lead to optical isomers (enantiomers).In contrast, the trans isomer, because of its symmetrical arrangement, retains a plane of symmetry, not allowing for optical isomers.Overall aspects of optical isomerism:

• Relates to the three-dimensional arrangement of atoms.
• Affects how complexes interact with polarized light.
• Is an essential consideration in many biochemical processes due to its impact on molecular interaction and recognition.